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Exponential decay in the frequency of analytic ranks of automorphic $$L$$-functions. (English) Zbl 1166.11326
From the text: This note should be seen as an addendum to the work of E. Kowalski and the second author [Duke Math. J. 100, 503–542 (1999; Zbl 1161.11359)] and deals with the problem of bounding, unconditionally, the order of vanishing at the critical point in a family of $$L$$-functions. This problem is illustrated in the particular case of $$L$$-functions of weight-2 primitive modular forms of prime level.
Recall the notation from that paper: For q a prime number, let $$S_2(q)^*$$ be the set of primitive forms of weight 2 and level $$q$$, normalized so that their first Fourier coefficient is 1; for $$f (z) =\sum_{n\geq 1} \lambda_f(n)n^{1/2}e(nz) \in S_2(q)^*$$, $$\lambda_f (1) = 1$$, let $$L(f, s) :=\sum_{n\geq 1}\lambda_f (n)n^{-s}$$, the associated (normalized) $$L$$-function; it admits analytic continuation to $$\mathbb C$$ with a functional equation relating $$L(f, s)$$ to $$L(f,1-s),$$ and we call $$r_f := \text{ord}_{s=1/2} L(f, s)$$ the analytic rank of $$f$$ .
In [Kowalski-Michel (loc. cit.)] the following was proved: There exists an absolute constant $$C_1 \geq 0$$ such that, for all $$q$$ prime, $$\sum_{f \in S_2(q)^*} r_f \leq C_1\,| S_2(q)^*|.$$
After that, much progress has been made concerning this question. In particular, in [E. Kowalski, P. Michel and J. M. VanderKam, J. Reine Angew. Math. 526, 1–34 (2000; Zbl 1020.11033)], a sharp explicit value was given for the constant $$C_1$$ ($$C_1 = 1.1891$$ for $$q$$ large enough). In the course of the proof, a uniform bound for the square of the ranks was obtained: $$\sum_{f \in S_2(q)^*}r_f ^2 \leq C_2\,| S_2(q)^*|.$$
However, the latter improvement used only a slight variant of the methods of Kowalski-Michel (loc. cit.). In fact, it is possible to pursue this idea further and it turns out that much more is true; this is the subject of the present note.
Consider a finite probability space $$(\Omega,\mu)$$, where $$\mu(\omega) > 0$$ for every $$\omega\in \Omega$$. For each $$\omega\in \Omega$$, suppose given a function $$h_\omega(s)$$ which is holomorphic in the half plane $$\text{Re}(s) \geq 0$$. Moreover assume the following hypothesis on the variance of the function $$h_\omega(s)-1$$:
Hypothesis: For some $$B,C > 0$$, $$M > 2$$, we have the bound
$\sum_{\omega\in \Omega}|h_\omega(\sigma + it) - 1|^2\mu(\omega) \leq C(1 + |t|)^BM^{-\sigma}$ uniformly for $$\sigma\geq (2 \log M)^{-1}$$.
Note that in view of this hypothesis and the fact that $$\mu(\omega) > 0$$, we have
$h_{\omega}(s) = 1 + O_{\omega}((1 + |t|)^{B/2}M^{-\sigma/2}) \tag{(*)}$
for each $$h_{\omega}$$. Thus $$h_{\omega}(s)$$ is nonvanishing for sufficiently large $$\sigma$$. For any $$\alpha\geq 0$$ and $$t_1 < t_2 \in \mathbb R$$, we may therefore define $$N(\omega, \alpha, t_1, t_2)$$ to be the number of zeros $$\rho$$ of $$h_{\omega}(s)$$ such that $$\text{Re}(\rho)\geq \alpha$$, $$t_1\leq \text{Im}(\rho)\leq t_2$$. Clearly $$N(\omega, \alpha, t_1, t_2)$$ is finite. Our general result gives an upper bound for the $$2k$$-th power of $$N(\omega, \alpha, t_1, t_2)$$ on average:
Theorem: With the above notations, assume that the above Hypothesis is satisfied. Then for all $$k\geq 1$$, for all $$\alpha\geq (\log M)^{-1}$$, and all $$t_1 < t_2,$$ we have
$\sum_{\omega\in \Omega} N(\omega, \alpha, t_1, t_2)^{2k}\mu(\omega)\ll C(k!)^2\left(48 \frac{k}{\alpha \log M}\right)^{2k} \left(1+|t|+\frac{16k}{\log M}\right)^BM^{-\alpha/2}(1+(t_2-t_1) \log M)$
where we have set $$|t| := \max(|t_1|, |t_2|)$$. The constant involved in the Vinogradov symbol is absolute.

##### MSC:
 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F30 Fourier coefficients of automorphic forms 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
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